CN111723512B - Determination method of axial symmetry dynamic response of pile foundation considering radial deformation influence - Google Patents

Determination method of axial symmetry dynamic response of pile foundation considering radial deformation influence Download PDF

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CN111723512B
CN111723512B CN202010644672.1A CN202010644672A CN111723512B CN 111723512 B CN111723512 B CN 111723512B CN 202010644672 A CN202010644672 A CN 202010644672A CN 111723512 B CN111723512 B CN 111723512B
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pile foundation
axial
displacement
pile
radial
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CN111723512A (en
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张石平
张军辉
林晨
徐站
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Changsha University of Science and Technology
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    • G06F30/00—Computer-aided design [CAD]
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Abstract

The invention discloses a method for determining axial symmetric dynamic response of a pile foundation in consideration of radial deformation influence, which respectively determines the radial displacement of the pile foundation caused by the side surface stress of the pile foundation, the axial displacement of the pile foundation caused by the side surface stress of the pile foundation, the radial displacement of the pile foundation caused by the bottom surface axial force of the pile foundation, the axial displacement of the pile foundation caused by the bottom surface axial force of the pile foundation, the radial displacement of the pile foundation caused by the bottom surface shear force of the pile foundation, the radial displacement of the pile foundation caused by the axial displacement of the top surface of the pile foundation under the action of an axial load and the axial displacement of the pile foundation caused by the axial displacement of the top surface of the pile foundation under the action of the axial load; and calculating the axial displacement and the radial displacement of the side surface of the pile foundation under the common action of the side surface stress of the pile foundation, the axial force of the bottom surface of the pile foundation, the shearing force of the bottom surface of the pile foundation and the axial displacement of the top surface of the pile foundation under the action of the axial load by adopting the determined displacement superposition. Meanwhile, axial and radial deformation of the pile foundation is considered, and the dynamic response of the obtained pile foundation is more in accordance with the actual engineering.

Description

Determination method of axial symmetry dynamic response of pile foundation considering radial deformation influence
Technical Field
The invention belongs to the technical field of civil engineering, and relates to a method for determining axial symmetry dynamic response of a pile foundation by considering radial deformation influence.
Background
The pile foundation is generally used in civil engineering, the research on the pile foundation generally regards the surrounding soil as an elastic medium, at the moment, the pile foundation and the soil form an interaction system, and the response of the pile foundation buried in the soil under the action of external load needs to be considered when the design or the research is carried out. The research on the dynamic response of the pile foundation is a theoretical basis in the engineering technical fields of pile foundation earthquake resistance, vibration reduction design, pile foundation dynamic detection and the like, and can provide guidance and reference for field construction. At present, the response of the pile foundation embedded in the elastic medium under the action of external force becomes a research hotspot in the field of applied mechanics. In civil engineering, such problems are directly related to the interaction analysis of the soil and structures such as pile foundations and anchors, which are commonly used in ground design and engineering practice (Scott, 1981). For such three-dimensional elastic theory problems, strict theoretical derivation is limited. For the axial symmetry problem under axial loading, a great deal of research has been carried out by researchers, such as Bose and Haldar (1985), Mylonakis and Gazetas (2002), Lu et al (2009), Seo et al (2009), Anoyatis et al (2013), Salgado et al (2013), Wu et al (2013), Naghibi et al (2014), Shadlou and Bhattacharya (2014), Hirai (2014), Zheng (2015), etc. In the above studies, the embedded pile foundation was assumed to be a one-dimensional structure. Therefore, the problem of the influence of the pile foundation radial deformation and bottom reaction force on the interaction with the surrounding medium has not been solved. However, the authors Pak and Gobert (1993), Masoumi et al (2007), and Masoumi and digrande (2008) have demonstrated in practical terms that we need to take into account the effects of the above factors by conducting careful studies on pile foundations that are subjected to axial loads and are fully embedded in an elastic medium. Although the basic rod theory is widely used in engineering practice due to its mathematical simplicity and practical value, unfortunately their use in this type of structure-continuum interaction problem may result in some fundamental drawbacks, particularly when the aspect ratio of the structure is not large enough. For example, axial and radial displacements of a buried element are generally dependent on tangential and lateral boundary forces exerted on it by the surrounding medium, whereas the basic rod theory can only describe axial deformations due to longitudinal loads. In fact, due to the poisson effect, there is also radial deformation and compression of the soil on the pile foundation, and it ignores radial deformation, so the basic rod theory essentially inhibits observation of proper lateral displacement and traction compatibility between the pile foundation and the soil. In addition to the above-mentioned non-physical risks, this approximation also poses serious limitations for correlation analysis of important issues, such as radial stress distribution and the influence of poisson's effect on the pile-soil system response (Pak and Gobert, 1993).
Disclosure of Invention
The embodiment of the invention aims to provide a method for determining axial symmetry dynamic response of a pile foundation in consideration of radial deformation influence, so as to solve the problems that the dynamic response result of the obtained pile foundation is inaccurate and does not accord with the actual engineering result because the pile foundation is assumed to be a one-dimensional rod structure and only the axial deformation of the pile foundation is considered in the conventional method for determining the dynamic response of the pile foundation.
The embodiment of the invention adopts the technical scheme that the method for determining the axial symmetry dynamic response of the pile foundation considering the radial deformation influence is carried out according to the following steps:
step S1, respectively determining radial displacement of the pile foundation caused by stress on the side surface of the pile foundation
Figure BDA0002572583520000021
Axial displacement of pile foundation caused by pile foundation side surface stress
Figure BDA0002572583520000022
Radial displacement of pile foundation caused by axial force of pile foundation bottom surface
Figure BDA0002572583520000023
Axial displacement of pile foundation caused by axial force of pile foundation bottom surface
Figure BDA0002572583520000024
Radial displacement of pile foundation caused by shear force of pile foundation bottom surface
Figure BDA0002572583520000025
Axial displacement of pile foundation caused by shear force of pile foundation bottom surface
Figure BDA0002572583520000026
Radial displacement of pile foundation caused by axial displacement of pile foundation top surface under axial load
Figure BDA0002572583520000027
Axial displacement of pile foundation caused by axial displacement of pile foundation top surface under axial load
Figure BDA0002572583520000028
Step S2, according to
Figure BDA0002572583520000029
And
Figure BDA00025725835200000210
and calculating the displacement of the pile foundation by adopting the following formula:
Figure BDA00025725835200000211
wherein u isr(z) represents the radial displacement of the side surface of the pile foundation under the common action of the stress of the side surface of the pile foundation, the axial force of the bottom surface of the pile foundation, the shearing force of the bottom surface of the pile foundation and the axial displacement of the top surface of the pile foundation under the action of the axial load; u. ofzAnd (z) represents the axial displacement of the side surface of the pile foundation under the common action of the stress of the side surface of the pile foundation, the axial force of the bottom surface of the pile foundation, the shearing force of the bottom surface of the pile foundation and the axial displacement of the top surface of the pile foundation under the action of the axial load, wherein z is more than or equal to 0 and less than or equal to l, and l is the length of the pile foundation.
The embodiment of the invention has the advantages that the axial and radial deformation of the pile foundation are considered at the same time, the non-torsion axisymmetric dynamic response instant harmonic axisymmetric response of the pile foundation is obtained through the superposition principle, the radial displacement and the axial displacement of the side surface of the pile foundation under the common action of the side surface stress of the pile foundation, the bottom surface axial force of the pile foundation, the bottom surface shearing force of the pile foundation and the axial displacement of the top surface of the pile foundation under the action of the axial load are obtained, the non-torsion axisymmetric response of the limited-length embedded pile foundation is more accurately obtained, and the dynamic response of the obtained pile foundation is more in line with the engineering practice. The method solves the problems that the dynamic response result of the pile foundation is inaccurate and does not conform to the actual engineering due to the fact that the pile foundation is assumed to be a one-dimensional rod structure and only the axial deformation of the pile foundation is considered in the conventional method for determining the dynamic response of the pile foundation.
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In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings used in the description of the embodiments or the prior art will be briefly described below, it is obvious that the drawings in the following description are only some embodiments of the present invention, and for those skilled in the art, other drawings can be obtained according to the drawings without creative efforts.
Fig. 1 is a schematic diagram of a mechanical model of a solid pile foundation under the action of an axisymmetric load.
Fig. 2 is an axisymmetric finite element model of the pile-soil system shown in fig. 1, which is created using the ADINA software.
Fig. 3 is a comparison graph of pile body displacement calculated by the method of the embodiment of the invention and a finite element method.
Fig. 4 is a diagram comparing the pile head speed calculated by the method of the embodiment of the invention and the finite element method.
Fig. 5 is a graph comparing the dynamic impedance of a two-dimensional rod piece in saturated soil with the dynamic impedance of a one-dimensional pile foundation.
Fig. 6 is a graph comparing the dynamic impedance of a two-dimensional rod piece in single-phase soil with the dynamic impedance of a one-dimensional pile foundation.
Fig. 7 is a graph of the effect of radial deformation of a pile in saturated soil on pile-to-soil system velocity response.
Fig. 8 is a graph of the effect of radial deformation of a pile in single phase soil on pile-to-soil system speed response.
Detailed Description
The technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the drawings in the embodiments of the present invention, and it is obvious that the described embodiments are only a part of the embodiments of the present invention, and not all of the embodiments. All other embodiments, which can be derived by a person skilled in the art from the embodiments given herein without making any creative effort, shall fall within the protection scope of the present invention.
FIG. 1 shows a linear elastic pile foundation according to an embodiment of the present invention, which has Lame constants of λ and μ, Poisson's ratio of ν, and radius a defined by radial stress t acting on the pile foundation side surfacerThe length l is represented by the axial stress t acting on the pile-side surfacezThe shear force of the top surface of the pile foundation is Q (0), the axial force of the top surface of the pile foundation is N (0), the shear force of the bottom surface of the pile foundation is Q (l), and the axial force of the bottom surface of the pile foundation is N (l). Under the load condition, the pile foundation is symmetrical about a z axis, all variables are independent of theta, namely the angular displacement of the pile foundation is ignored, the embodiment of the invention is a cylindrical coordinate system, the origin of coordinates is located at the top center of the pile foundation, the z axis direction is downward, and theta corresponds to z and r.
It is helpful to recognize that the main deformations of axially loaded embedded pilings with sufficient slenderness ratio and stiffness are axial compression and longitudinal displacement, and in view of this it seems logical to represent the axial displacement field of the pilings with a first order approximation, i.e. it is logicalWhen the slenderness ratio and the rigidity of the pile foundation are large enough, the radial displacement of the pile foundation is small under the action of axial load, in this case, it is reasonable to adopt first-order approximation to represent the axial displacement of the pile foundation, and the displacement of the pile foundation is only a function of a height variable z at the moment. However, when the pile foundation is subjected to axial loads, a corresponding radial displacement field will generally also occur due to the poisson effect. In addition, the pilings are laterally constrained by the surrounding soil mass and are likely to be subjected to significant internal radial compression due to the boundary lateral stresses caused by the load acting on their circumferential surfaces. As a preliminary attempt to take these physical aspects into account without introducing unnecessary complexity, a first non-trivial approximation of the change in axial displacement of the pile foundation in the radial direction across its cross-section was adopted (Pak and Gobert, 1993). In addition to considering the radial displacement caused by the poisson effect and the radial compression effect of the surrounding soil mass on the pile foundation as described above, these kinematic assumptions have also been shown to model the radial shear phenomena, which are important in the wave propagation problem (Mindlin and Herrmann, 1951). Therefore, first, u is adoptedr(z) radial displacement at pile foundation side height z, using uz(z) represents the axial displacement at pile base side height z, then:
Figure BDA0002572583520000041
wherein u isr(r, z) represents the radial displacement of the pile at radius r and height z, uzAnd (r, z) represents the axial displacement of the pile foundation at the radius r and the height z, wherein r is more than or equal to 0 and less than or equal to a, and z is more than or equal to 0 and less than or equal to l.
Then, the axial force and the shear force of the cross section of the pile foundation are defined as follows:
Figure BDA0002572583520000042
wherein σzzRepresenting the positive stress, σ, acting in the z-plane and in the z-directionzrDenotes shear stress acting on the z-plane and in the r-direction, dS denotesIntegrating the area, namely integrating the axial stress and the tangential stress on the cross section area of the pile foundation at the height z to obtain the axial force and the shearing force of the cross section of the pile foundation, wherein A represents the area of the cross section of the pile foundation, and A is pi a2(ii) a J represents the polar moment of inertia of the pile foundation,
Figure BDA0002572583520000043
the elastic potential energy of the pile foundation is as follows:
Figure BDA0002572583520000044
wherein σ is stress tensor, ε is strain tensor, V is volume, σrrRepresenting a positive stress, epsilon, acting on the r-plane and in the r-directionrrRepresents the positive strain acting on the r-plane and in the radial direction; sigmarzRepresenting the shear stress, epsilon, acting in the r-plane and in the z-directionrzRepresenting the shear strain, σ, acting in the r-plane and in the z-directionθθRepresenting a positive stress, ε, acting in the θ plane and in the θ directionθθRepresenting positive strain, ε, acting in the θ plane and in the θ directionzzRepresenting a positive strain acting in the z-plane and in the z-direction.
The kinetic energy of the pile is expressed as:
Figure BDA0002572583520000051
wherein p represents the density of the pile foundation body, v represents the velocity vector of the pile foundation, namely the velocity vector is composed of radial velocity components and vertical velocity components,
Figure BDA0002572583520000052
is uz(z) first derivative with respect to time t.
Because the pile foundation receives the effect of surrounding medium and pile foundation base force, its nonconservative power does work and does:
Figure BDA0002572583520000053
wherein t is the external force vector that whole pile foundation surface received, and the direction of different external forces is different, and the atress condition on pile foundation surface has been marked in figure 1. u denotes the displacement vector of the pile foundation, ur(0) Indicating radial displacement of pile top surface, ur(l) Indicating radial displacement of pile base surface, uz(0) Indicating axial displacement of pile top surface, uz(l) Indicating axial displacement of the pile base.
The virtual work of the external load can be expressed as (Morse and Feshbach, 1953):
Figure BDA0002572583520000054
wherein, delta is a variation symbol, delta ur(z) represents urVariation of (z), δ uz(z) represents uz(z) variation.
According to Hamilton's kinetic principle (Achenbach, 1973):
Figure BDA0002572583520000055
wherein T represents the kinetic energy of the pile foundation, P represents the elastic potential energy of the pile foundation, and T1、t2Two different time points are randomly selected in the movement process of the pile foundation.
1-7, deducing the motion equation of the pile foundation as follows:
Figure BDA0002572583520000056
wherein the content of the first and second substances,
Figure BDA0002572583520000057
is uz(z) second derivative with respect to t.
Since the current work considers the factor e over timeiωtThe changing steady state vibration, equation (8) can be further written as:
Figure BDA0002572583520000061
wherein f isr(z) denotes the radial annular load, fz(z) represents the axial annular load,
Figure BDA0002572583520000062
Figure BDA0002572583520000063
since each item contains a time factor eiωtFor convenience of expression, the time factor e is used in the analysis processiωtAre omitted.
The homogeneous equation of equation (9) is:
Figure BDA0002572583520000064
let ur(z)＝Ureηz，uz(z)＝UzeηzSubstituting the change law into a formula (10), namely a radial and longitudinal vibration change law u of the pile foundationr(z) and uz(z) making assumptions, converting the differential equation into an algebraic equation, and obtaining:
Figure BDA0002572583520000065
wherein, UrIndicating the radial displacement amplitude, U, of the pile foundationzExpressing the axial displacement amplitude of the pile foundation, wherein eta is a constant to be solved, and after solving eta, the formula (11) is changed into two linear equations which can be solved into UrAnd UzFurther solve for ur(z) and uz(z)。
If equation (11) has a non-trivial solution, requiring the determinant of the coefficient matrix of equation (11) to be 0, one can obtain:
Figure BDA0002572583520000066
wherein the content of the first and second substances,
Figure BDA0002572583520000067
namely, it is
Figure BDA0002572583520000068
By kpFor expression
Figure BDA0002572583520000069
The substitution is made to simplify the formula.
The roots of the formula (12) are each η1＝η2＝0，η3＝kp，η4＝-kp. Thus, a homogeneous solution of equation (9) can be written:
Figure BDA0002572583520000071
wherein h isj(z) is a function of the variation of the displacement of the pile in the radial direction, bj(z) is a function of the axial variation of the displacement of the pile foundation, DjIs a constant that can be determined from boundary conditions, where:
Figure BDA0002572583520000072
as can be seen from fig. 1, the boundary conditions of the displacement solution of the pile foundation considering axial and radial deformation under the action of the axial simple harmonic load can be represented as follows:
on the top surface of the solid pile foundation:
uz(z)|z＝0＝Δz； (15)
wherein, DeltazThe axial displacement of the top surface of the pile foundation under the action of axial load.
The surface conditions without friction were:
Q(z)|z＝0＝0； (16)；
wherein q (z) represents the shear force of the pile foundation.
The boundary conditions at the bottom of the solid pile foundation pile are as follows:
N(z)|z＝l＝N(l)； (17)
Q(z)|z＝l＝Q(l)； (18)
where n (z) represents the axial force of the pile foundation.
According to the principle of superposition, the displacement of the pile foundation can be written as:
Figure BDA0002572583520000073
wherein u isr(z) represents the radial displacement of the side surface of the pile foundation under the common action of the stress of the side surface of the pile foundation, the axial force of the bottom surface of the pile foundation, the shearing force of the bottom surface of the pile foundation and the axial displacement of the top surface of the pile foundation under the action of the axial load; u. ofz(z) axial displacement of the side surface of the pile foundation under the common action of the stress of the side surface of the pile foundation, the axial force of the bottom surface of the pile foundation, the shear force of the bottom surface of the pile foundation and the axial displacement of the top surface of the pile foundation under the action of the axial load;
Figure BDA0002572583520000081
radial displacement of the pile caused by stress on the side surface of the pile,
Figure BDA0002572583520000082
axial displacement of the pile caused by stress on the side surface of the pile,
Figure BDA0002572583520000083
the radial displacement caused by the axial force of the bottom surface of the pile foundation,
Figure BDA0002572583520000084
is axial displacement caused by axial force of the bottom surface of the pile foundation,
Figure BDA0002572583520000085
is radial displacement caused by the shearing force of the bottom surface of the pile foundation,
Figure BDA0002572583520000086
for shear initiation of pile foundation bottom surfaceIs moved in the axial direction of the shaft,
Figure BDA0002572583520000087
the radial displacement of the pile foundation caused by the axial displacement of the top surface of the pile foundation under the action of the axial load,
Figure BDA0002572583520000088
the axial displacement of the pile foundation caused by the axial displacement of the top surface of the pile foundation under the action of the axial load.
Figure BDA0002572583520000089
And
Figure BDA00025725835200000810
t being non-zerorAnd tzAnd n (l) ═ q (l) ═ 0 and ΔzDisplacement obtained when 0;
Figure BDA00025725835200000811
and
Figure BDA00025725835200000812
non-zero N (l) and Q (l) 0, tr＝tz＝0、ΔzDisplacement obtained when 0;
Figure BDA00025725835200000813
and
Figure BDA00025725835200000814
non-zero Q (l) and N (l) 0, tr＝tz＝0、ΔzDisplacement obtained when 0;
Figure BDA00025725835200000815
and
Figure BDA00025725835200000816
a non-zerozAnd n (l) ═ q (l) ═ 0, tr＝tzDisplacement obtained when 0.
Solving for
Figure BDA00025725835200000817
And
Figure BDA00025725835200000818
to obtain
Figure BDA00025725835200000819
And
Figure BDA00025725835200000820
firstly, a pair of unit annular loads f acting in the area of s being more than or equal to 0 and less than or equal to l is obtainedrAnd fzGreen function of, i.e. annular load f of pile foundation in radial directionrAnd axial annular load fzUnder the action of the Green function, the unit annular load refers to the load of unit size borne by the annular side surface of the pile foundation under the unit length.
(1) Vertical ring load f at z ═ s, r ═ azRadial annular load f with a sum of zerorCan be expressed as:
Figure BDA00025725835200000821
where δ () represents a dirac function.
Along the position of the annular load, the pile foundation is divided into two parts, namely a region I with 0 & ltz & lt s and a region II with s & ltz & lt l, s represents the position of the axial or radial annular load action and corresponds to a known quantity, and z represents the position of any point on the pile foundation, so that the solution of the two parts of the formula (13) can be expressed as follows:
in region I where z is greater than or equal to 0 and less than s:
Figure BDA00025725835200000822
in the region s < z ≦ l, i.e. region II:
Figure BDA0002572583520000091
wherein the content of the first and second substances,
Figure BDA0002572583520000092
represents the radial displacement of the pile foundation in the area I when the radial annular load is 0 and the axial annular load is not 0,
Figure BDA0002572583520000093
the axial displacement of the pile foundation in the area I when the radial annular load is 0 and the axial annular load is not 0 is represented;
Figure BDA0002572583520000094
is a constant of the displacement of the pile foundation when the radial annular load is 0 and the axial annular load is not 0,
Figure BDA0002572583520000095
is a constant of pile foundation displacement in a region I, namely z is more than or equal to 0 and less than s when the radial annular load is 0 and the axial annular load is not 0,
Figure BDA0002572583520000096
the constant of the displacement of the pile foundation in the area II, i.e. the area where s is more than z and less than or equal to l when the radial annular load is 0 and the axial annular load is not 0;
Figure BDA0002572583520000097
represents the radial displacement of the pile foundation in the area II when the radial annular load is 0 and the axial annular load is not 0,
Figure BDA0002572583520000098
and represents the axial displacement of the pile foundation in the area II when the radial annular load is 0 and the axial annular load is not 0.
Figure BDA0002572583520000099
And
Figure BDA00025725835200000910
are unknown 8 constants whose values depend on the pile boundaries of the aforementioned assumptionsConditions and displacement and force continuity at the interface of zone I, zone II.
At pile foundation top surface z-0 and pile base z-l there are:
Figure BDA00025725835200000911
wherein Q isZI(z) represents the shearing force of the area I which is the top section of the pile foundation when the radial annular load is 0 and the axial annular load is not 0, QZIIAnd (z) represents the shearing force of the pile foundation bottom section, namely the area II when the radial annular load is 0 and the axial annular load is not 0.
At the position where the section z of the pile body is equal to s, the displacement continuity condition is as follows:
Figure BDA00025725835200000912
wherein s is+Denotes an infinite approximation of s from the negative z-axis-Representing a positive infinite proximity to s from the z-axis, i.e. s+Refers to the lower surface of the pile foundation at a depth s, s-Refers to the upper surface of the pile foundation with the depth of s,
Figure BDA00025725835200000913
s in region II when the radial annular load is 0 and the axial annular load is not 0+The radial displacement of the (c) axis,
Figure BDA00025725835200000914
s represents the pile foundation in the region I when the radial annular load is 0 and the axial annular load is not 0-The radial displacement of the (c) axis,
Figure BDA0002572583520000101
s represents the pile foundation in the region I when the radial annular load is 0 and the axial annular load is not 0-The axial displacement of the (c) is,
Figure BDA0002572583520000102
representing radial annular loadsS in region II for pile foundations with axial annular load not 0+Axial displacement of (a).
The displacement gradient is continuous, i.e.:
Figure BDA0002572583520000103
the equilibrium conditions, namely:
NZI(s-；s)-NZII(s+；s)＝2πa； (26)
wherein N isZI(s-(ii) a s) represents the pile foundation in the area I when the radial annular load is 0 and the axial annular load is not 0-Axial force of (C), NZII(s+(ii) a s) represents the pile foundation in the area II when the radial annular load is 0 and the axial annular load is not 0+Axial force of (d).
By substituting equations (21) to (22) into equations (23) to (26), unknown constants can be obtained
Figure BDA0002572583520000104
And
Figure BDA0002572583520000105
the Green function for the presence of only vertical annular load and a radial annular load of 0 is obtained as:
Figure BDA0002572583520000106
wherein the content of the first and second substances,
Figure BDA0002572583520000107
indicating pile foundation in vertical annular load fzThe radial displacement under the action of the Green function,
Figure BDA0002572583520000108
indicating pile foundation in vertical annular load fzAn axial displacement Green function under action;
Figure BDA0002572583520000109
and is
Figure BDA00025725835200001010
Figure BDA00025725835200001011
Figure BDA00025725835200001012
Figure BDA00025725835200001013
(2) Radial annular load f at z ═ s, r ═ arVertical annular load f with a sum of zerozCan be expressed as:
Figure BDA00025725835200001014
where δ () represents a dirac function.
Along the position of the annular load, the pile foundation is divided into two parts, namely a region I with 0 & lt z & lt s and a region II with s & lt z & lt l, so that the solution of the two parts of the formula (13) can be expressed as follows:
in region I where z is greater than or equal to 0 and less than s:
Figure BDA0002572583520000111
in the region s < z ≦ l, i.e. region II:
Figure BDA0002572583520000112
wherein the content of the first and second substances,
Figure BDA0002572583520000113
the radial displacement of the pile foundation in the area I when the axial annular load is 0 and the radial annular load is not 0,
Figure BDA0002572583520000114
the axial displacement of the pile foundation in the area I when the axial annular load is 0 and the radial annular load is not 0,
Figure BDA0002572583520000115
the radial displacement of the pile foundation in the area II when the axial annular load is 0 and the radial annular load is not 0,
Figure BDA0002572583520000116
and the axial displacement of the pile foundation in the area II is shown when the axial annular load is 0 and the radial annular load is not 0.
Figure BDA0002572583520000117
Is a constant of the displacement of the pile foundation when the axial annular load is 0 and the radial annular load is not 0,
Figure BDA0002572583520000118
is a constant of pile foundation displacement in a region I, namely z is more than or equal to 0 and less than s when the axial annular load is 0 and the radial annular load is not 0,
Figure BDA0002572583520000119
the constant of the displacement of the pile foundation in the area II, i.e. the area where s is more than z and less than or equal to l when the axial annular load is 0 and the radial annular load is not 0. Wherein the content of the first and second substances,
Figure BDA00025725835200001110
and
Figure BDA00025725835200001111
are unknown 8 constants whose values depend on boundary conditions and continuity conditions.
At pile foundation top surface z-0 and pile base z-l there are:
Figure BDA00025725835200001112
wherein Q isRI(z) is a radial ring with axial annular load of 0Shear force of area I, i.e. pile top section, Q when the form load is not 0RIIAnd (z) is the shearing force of the bottom surface of the pile foundation in the area II when the axial annular load is 0 and the radial annular load is not 0.
The section z of the pile body is as s:
displacement continuous conditions, namely:
Figure BDA0002572583520000121
wherein the content of the first and second substances,
Figure BDA0002572583520000122
s represents the pile foundation in the region I when the axial annular load is 0 and the radial annular load is not 0-The radial displacement of the (c) axis,
Figure BDA0002572583520000123
indicates that the pile foundation is in the area II s when the axial annular load is 0 and the radial annular load is not 0+A radial displacement of (a);
Figure BDA0002572583520000124
s represents the pile foundation in the region I when the axial annular load is 0 and the radial annular load is not 0-The axial displacement of the (c) is,
Figure BDA0002572583520000125
indicates that the pile foundation is in the area II s when the axial annular load is 0 and the radial annular load is not 0+Axial displacement of (a).
The displacement gradient condition, namely:
Figure BDA0002572583520000126
the equilibrium conditions, namely:
QRI(s-；s)-QRII(s+；s)＝2πa； (34)
wherein Q isRI(s-(ii) a s) represents an axial annular load of 0, diameterS in region I when annular load is not 0-Axial force of (Q)RII(s+(ii) a s) represents the pile foundation in the area II when the axial annular load is 0 and the radial annular load is not 0+Axial force of (d).
By substituting equations (29) to (30) into equations (31) to (34), unknown constants can be obtained, and the Green function of the pile foundation when the axial annular load is 0 and the radial annular load is not 0 is obtained as:
Figure BDA0002572583520000127
wherein the content of the first and second substances,
Figure BDA0002572583520000128
indicating radial annular load f of pile foundationrThe radial displacement under the action of the Green function,
Figure BDA0002572583520000129
indicating radial annular load f of pile foundationrThe Green function of the axial displacement under the action,
Figure BDA00025725835200001210
and is
Figure BDA00025725835200001211
Figure BDA00025725835200001212
Figure BDA0002572583520000131
Figure BDA0002572583520000132
By the expressions (27) and (35), displacement
Figure BDA0002572583520000133
And
Figure BDA0002572583520000134
can be written as:
Figure BDA0002572583520000135
Figure BDA0002572583520000136
wherein f isr(s) denotes the radial annular load at s, fz(s) represents the axial annular load at s;
Figure BDA0002572583520000137
representing the radial displacement caused by the pile side surface stress at s,
Figure BDA0002572583520000138
representing axial displacement caused by pile foundation side surface stress at s; t is tr(s) represents the radial stress of the pile-side surface at s, tz(s) represents axial stress of the pile side surface at s;
Figure BDA0002572583520000139
indicating pile foundation in fr(s) a Green function of the radial displacement at z,
Figure BDA00025725835200001310
indicating pile foundation in fr(s) an axial displacement Green function at z under the influence of;
Figure BDA00025725835200001311
indicating pile foundation in fz(s) a Green function of the radial displacement at z,
Figure BDA00025725835200001312
indicating pile foundation in fz(s) the Green function of the axial displacement at z,
Figure BDA00025725835200001313
and
Figure BDA00025725835200001314
the calculation is carried out by the formula (27),
Figure BDA00025725835200001315
and
Figure BDA00025725835200001316
calculated by equation (35), i.e.:
Figure BDA00025725835200001317
Figure BDA00025725835200001318
Figure BDA00025725835200001319
Figure BDA00025725835200001320
Figure BDA00025725835200001321
and
Figure BDA00025725835200001322
essentially, it is
Figure BDA00025725835200001323
And
Figure BDA00025725835200001324
the algebraic equations can be obtained by numerically discrete decomposition of the integral numbers, and then the right unknowns are moved to the left and combined to solve for (36) and (37). t is tr(s) and tz(s) can be obtained by reference to existing literature, e.g.Pak and Gobert (1993)); or model calculation is established through finite element software or the actual monitoring of the pile foundation surface is carried out to obtain discrete data points, and then data fitting is carried out to obtain a specific stress expression as tr(s) and tz(s)。
Derivation of
Figure BDA0002572583520000141
And
Figure BDA0002572583520000142
response of pile foundation according to stacking principle
Figure BDA0002572583520000143
And
Figure BDA0002572583520000144
can be decomposed into two parts, namely:
Figure BDA0002572583520000145
wherein the content of the first and second substances,
Figure BDA0002572583520000146
represents the radial static displacement of the pile foundation caused by N (l),
Figure BDA0002572583520000147
the axial static displacement of the pile foundation caused by N (l) is shown,
Figure BDA0002572583520000148
representing the radial dynamic displacement caused by the inertial term,
Figure BDA0002572583520000149
representing the axial dynamic displacement caused by the inertial term. In the present embodiment, N (l) is considered to be a known quantity, and can be obtained by referring to the existing research literature (e.g., Pak and Gobert (1993)), or by finite element modeling calculation or actual monitoring。
(1) Solving for
Figure BDA00025725835200001410
And
Figure BDA00025725835200001411
at this time, it is assumed that only the pile foundation bottom surface axial force n (l) exists, and the static displacement of the pile foundation corresponds to the homogeneous form of the motion equation, so that the motion equation of the pile foundation is as follows:
Figure BDA00025725835200001412
the boundary conditions are as follows:
Figure BDA00025725835200001413
wherein the content of the first and second substances,
Figure BDA00025725835200001414
showing the corresponding shearing force of the static displacement part under the action of the axial force N (l) of the bottom surface of the pile foundation,
Figure BDA00025725835200001415
the axial force corresponding to the static displacement part of the pile foundation under the action of the axial force N (l) of the bottom surface of the pile foundation is shown.
Bringing formula (13) into formula (40) yields:
Figure BDA0002572583520000151
(2) solving for
Figure BDA0002572583520000152
And
Figure BDA0002572583520000153
because the inertia term is contained and corresponds to a non-homogeneous motion equation, the motion equation of the pile foundation is as follows:
Figure BDA0002572583520000154
the boundary conditions are as follows:
Figure BDA0002572583520000155
wherein the content of the first and second substances,
Figure BDA0002572583520000156
represents the corresponding shearing force of the dynamic displacement part of the pile foundation under the action of the inertia term,
Figure BDA0002572583520000157
and the axial force corresponding to the dynamic displacement part of the pile foundation under the action of the inertia term is represented.
Since the equations (27) and (35) are Green functions under the action of unit annular load, which can be understood as a general solution of the heterogeneous equation under the action of unit inertia terms, the influence of the right heterogeneous term is further considered here, and is integrated in the whole length range of the pile foundation, so as to expand into a displacement function of the pile foundation at any depth, and therefore, the solution of the equation (42) is obtained through the equations (27) and (35):
Figure BDA0002572583520000158
Figure BDA0002572583520000159
wherein the content of the first and second substances,
Figure BDA00025725835200001510
the radial static displacement of the pile foundation at s caused by the axial force N (l) of the bottom surface of the pile foundation,
Figure BDA00025725835200001511
for radial dynamic displacement of the pile foundation at s caused by the inertial term,
Figure BDA00025725835200001512
the axial static displacement of the pile foundation at s caused by the axial force N (l) of the bottom surface of the pile foundation,
Figure BDA00025725835200001513
is the axial dynamic displacement of the pile foundation at s caused by the inertia term. The equations (44) and (45) can be solved by performing numerical discrete decomposition on the integral numbers to obtain an algebraic equation system, and then moving the right unknown quantity to the left for combination.
Derivation of
Figure BDA0002572583520000161
And
Figure BDA0002572583520000162
similarly, the pile response according to the superposition principle
Figure BDA0002572583520000163
And
Figure BDA0002572583520000164
can be broken down into two parts, namely:
Figure BDA0002572583520000165
wherein the content of the first and second substances,
Figure BDA0002572583520000166
for radial static displacement of the pile foundation caused by pile foundation floor shear forces q (l),
Figure BDA0002572583520000167
the axial static displacement of the pile foundation caused by the pile foundation bottom surface shearing force Q (l),
Figure BDA0002572583520000168
for radial dynamic displacement due to the inertial term,
Figure BDA0002572583520000169
is the axial dynamic displacement caused by the inertial term.
(1) Solving for
Figure BDA00025725835200001610
And
Figure BDA00025725835200001611
the displacement at this moment is only axial displacement and radial displacement generated by shearing force of the bottom surface of the pile foundation, so that the displacement is substituted into the motion equation, and the motion equation of the pile foundation at this moment is obtained as follows:
Figure BDA00025725835200001612
the boundary conditions are as follows:
Figure BDA00025725835200001613
wherein the content of the first and second substances,
Figure BDA00025725835200001614
the shearing force corresponding to the static displacement part of the pile foundation under the action of the shearing force Q (l) of the bottom surface of the pile foundation,
Figure BDA00025725835200001615
the axial force is corresponding to the static displacement part of the pile under the action of the shearing force Q (l) of the bottom surface of the pile.
Substituting equation (13) into equation (48) yields:
Figure BDA00025725835200001616
(2) solving for
Figure BDA00025725835200001617
And
Figure BDA00025725835200001618
the dynamic displacement caused by the inertia term generates a non-homogeneous term in a motion equation, and the motion equation of the pile foundation at the moment is as follows:
Figure BDA0002572583520000171
the boundary conditions are as follows:
Figure BDA0002572583520000172
wherein the content of the first and second substances,
Figure BDA0002572583520000173
is the shearing force corresponding to the dynamic displacement part of the pile foundation,
Figure BDA0002572583520000174
the axial force is corresponding to the dynamic displacement part of the pile foundation.
Equations (27) and (35) are Green functions of unit vertical and radial annular loads, and can form a general solution to the heterogeneous equation, so that the solution of equation (50) is found by equations (27) and (35):
Figure BDA0002572583520000175
Figure BDA0002572583520000176
wherein the content of the first and second substances,
Figure BDA0002572583520000177
the radial static displacement of the pile foundation at s caused by the shearing force Q (l) of the bottom surface of the pile foundation,
Figure BDA0002572583520000178
for the radial dynamic displacement of the pile foundation at s caused by the inertia term,
Figure BDA0002572583520000179
the axial static displacement of the pile foundation at s caused by the shearing force Q (l) of the bottom surface of the pile foundation,
Figure BDA00025725835200001710
is the axial dynamic displacement of the pile foundation at s caused by the inertia term. Equations (52) and (53) can be solved by performing numerical discrete decomposition on the integral numbers to obtain an algebraic equation system, and then moving the right unknown quantities to the left for combination. In the present embodiment, Q (l) is considered to be a known quantity, and can be obtained by consulting the existing research literature (such as Pak and Gobert (1993)), or by finite element modeling calculation or actual monitoring.
Derivation of
Figure BDA00025725835200001711
And
Figure BDA00025725835200001712
response of pile foundation according to stacking principle
Figure BDA00025725835200001713
And
Figure BDA00025725835200001714
can be divided into two parts, i.e.
Figure BDA00025725835200001715
Wherein the content of the first and second substances,
Figure BDA00025725835200001716
is formed by axial displacement delta of pile foundation top surface under the action of axial loadzInduced radial staticsThe displacement is carried out in such a way that,
Figure BDA00025725835200001717
is formed by axial displacement delta of pile foundation top surface under the action of axial loadzThe resulting axial static displacement is caused by the axial displacement,
Figure BDA00025725835200001718
is the radial dynamic displacement caused by the inertial term,
Figure BDA0002572583520000181
is the axial dynamic displacement caused by the inertial term.
(1) Solving for
Figure BDA0002572583520000182
And
Figure BDA0002572583520000183
at the moment, the axial displacement delta of the pile foundation top surface under the action of axial load only existszThe static displacement of the pile foundation corresponds to the homogeneous form of the motion equation, so the motion equation of the pile foundation at the moment is as follows:
Figure BDA0002572583520000184
the boundary conditions are as follows:
Figure BDA0002572583520000185
wherein the content of the first and second substances,
Figure BDA0002572583520000186
showing axial displacement delta of pile foundation top surface under axial loadzThe shear force of the pile foundation corresponding to the static displacement of the pile foundation is caused,
Figure BDA0002572583520000187
indicating pile foundation top surface under axial loadAxial displacement ofzThe static displacement of the pile foundation caused corresponds to the axial force of the pile foundation.
By the formulae (13) and (56), we obtain:
Figure BDA0002572583520000188
(2) solving for
Figure BDA0002572583520000189
And
Figure BDA00025725835200001810
the dynamic displacement caused by the inertia term generates a non-homogeneous term in the motion equation, so that the motion equation of the pile foundation is as follows:
Figure BDA00025725835200001811
the boundary conditions are as follows:
Figure BDA0002572583520000191
wherein the content of the first and second substances,
Figure BDA0002572583520000192
showing axial displacement delta of pile foundation top surface under axial loadzThe shearing force of the pile foundation corresponding to the dynamic displacement of the pile foundation is caused,
Figure BDA0002572583520000193
showing axial displacement delta of pile foundation top surface under axial loadzThe dynamic displacement of the pile foundation caused by the dynamic displacement corresponds to the axial force of the pile foundation.
Equations (27) and (35) are Green functions of the unit annular load, and can constitute a general solution to the heterogeneous equation, so that the solution of equation (58) is obtained by equations (27) and (35):
Figure BDA0002572583520000194
Figure BDA0002572583520000195
wherein the content of the first and second substances,
Figure BDA0002572583520000196
is axial displacement delta of pile foundation top surface under the action of axial loadzThe radial dynamic displacement of the pile foundation at the position s is caused,
Figure BDA0002572583520000197
is axial displacement delta of pile foundation top surface under the action of axial loadzAnd (4) causing axial dynamic displacement of the pile foundation at the position s. Examples of the inventionzConsidered a known quantity, can be obtained by monitoring. The equations (60) and (61) can be solved by performing numerical discrete decomposition on the integral numbers to obtain an algebraic equation system, and then moving the right unknown quantity to the left for combination.
Finally, it follows from equation (19):
Figure BDA0002572583520000198
Figure BDA0002572583520000199
wherein, the Lame constants lambda and mu are calculated according to the elastic modulus and the Poisson ratio, and the specific calculation formula is as follows: λ ═ υ E/[ (1+ υ) (1-2 υ) ], μ ═ E/2(1+ υ), E represents the elastic modulus of the pile foundation, the pile foundation density ρ is generally given by a design unit and can be detected by field tests, and the vibration frequency ω of the pile foundation depends on the frequency of an applied load.
Wherein z is more than or equal to 0 and less than or equal to l, and:
Figure BDA0002572583520000201
wherein the content of the first and second substances,
Figure BDA0002572583520000202
representing the static radial displacement function under the action of the unit pile foundation bottom surface axial force N (l),
Figure BDA0002572583520000203
representing a static axial displacement function under the action of the unit pile foundation bottom surface axial force N (l);
Figure BDA0002572583520000204
representing the static radial displacement function under the action of the shearing force Q (l) of the bottom surface of the unit pile foundation,
Figure BDA0002572583520000205
and (3) representing a static axial displacement function under the action of the shearing force Q (l) of the bottom surface of the unit pile foundation.
The dynamic response comprises displacement, stress, internal force, dynamic impedance, speed response and the like, and the embodiment of the invention mainly provides a displacement expression of the pile foundation, so that the expressions of the stress, the internal force, the dynamic impedance, the speed response and the like can be obtained by deducing the displacement of the pile foundation. Finally u is given by formula (19)r(z) and uzThe calculation formula of (z) can be widely applied to interaction analysis of the pile foundation-soil system under the action of the axial load. It should be noted that, in the actual engineering, the pile diameter, the pile foundation length, the pile body elastic modulus, the poisson ratio, the pile body density, the harmonic excitation force frequency, which are generally known material and geometric parameters, are taken into the equations (62) and (63) as basic parameters to be calculated. If not known, can also be obtained by experimental tests. Furthermore, in order to calculate pile foundation displacement, it is also necessary to know the radial and vertical stresses on the pile foundation side, i.e. trAnd tz. Firstly, determining the radial acting force t of the surrounding soil body to the pile foundationrAnd a vertical force tzThe stress values can be obtained by testing stress sensors embedded on the pile body, for example, after the stress values of some measuring points along the pile body are obtained, data fitting is carried outObtaining a mathematical formula of the radial and vertical stress of the pile foundation side along the vertical coordinate z, or directly obtaining a formula of the radial and vertical stress of the pile foundation side by consulting the existing literature (such as the pile foundation side soil body stress proposed by the literature Pak and Gobert (1993), and then obtaining a formula t of the pile foundation side stressrAnd tzAnd carrying out simultaneous calculation in the formulas (62) and (63) to obtain the radial and vertical displacements of the pile foundation. And then further obtaining physical quantities which are interested or need to be considered according to specific needs, such as pile top vertical dynamic impedance (dividing pile top vertical acting force by pile top vertical displacement) or speed response (deriving displacement with time) and the like. In other words, it is the core to find the radial and vertical displacements of the pile foundation, and although we only use the vertical displacement for research or application in the subsequent analysis, the vertical displacement is obtained under the condition of considering the radial deformation of the pile foundation, which is more practical, that is, the significance of the invention is.
The vibration characteristics of the pile foundation need to be considered when the work such as the anti-seismic and vibration-damping design and the pile foundation power detection of the pile foundation is carried out, the vibration characteristics of the pile foundation are deeply analyzed based on the power response of the pile foundation, the influence of different design parameters (such as pile foundation length, pile foundation diameter, pile foundation body modulus and the like) on the characteristics such as the pile foundation power impedance, pile foundation body speed and reflected waves is researched, and then reference is provided for the pile foundation anti-seismic and vibration-damping design and the pile foundation power detection. The dynamic response of the pile foundation is accurately calculated, and the anti-seismic and vibration-damping design of the pile foundation can be optimized. At present, due to the complexity of the problem of the dynamic interaction of the soil-pile foundation, a plurality of assumptions are adopted to simplify the analysis of the problem, but the assumptions also cause certain difference between the theory of design and detection and the actual engineering, so that a plurality of designers tend to be conservative when carrying out the dynamic design of the pile foundation to cause economic waste, and errors can be caused to the detection result, therefore, the dynamic response of the pile foundation is accurately calculated, the anti-seismic and vibration-damping design of the pile foundation can be optimized, the economic benefit of the engineering is improved, and the accuracy of the detection result is improved.
Numerical example verification:
in order to verify the correctness of the method provided by the embodiment of the invention, an axis-symmetric finite element model of the pile foundation-soil system in fig. 1 is established by using the ADINA software, and the model is shown in fig. 2. The dynamic response obtained by the method of the embodiment of the invention is compared with the pile foundation body displacement and pile foundation head speed calculated by a finite element method, and the result is shown in fig. 3-4. In the finite element model, the soil is assumed to be porous material, and the soil body is simulated by using 9-node rectangular units. The infinite boundary conditions were simulated by setting the left side of the model as an axisymmetric boundary, the surface of the soil layer as a free boundary, the bottom of the model as a watertight fixed boundary, and the right side as a fixed boundary without pore pressure, so as to be consistent with the boundary conditions specified in fig. 1. It is worth noting that in this example, a model width of 50m has been used to obtain the steady state response amplitude, and the right side boundary effect is better eliminated. As can be seen from the comparison of FIGS. 3-4, the dynamic response solution of the embodiment of the invention can be well matched with the finite element simulation result, thereby verifying that the method is correct.
The dynamic impedance (the ratio of the external force on the top of the pile foundation divided by the displacement of the top of the pile foundation, i.e., the axial acting force loaded on the pile foundation) of the two-dimensional rod obtained in the embodiment of the present invention is compared with the dynamic impedance of the one-dimensional pile foundation obtained by Liu (2014), etc., and the results are shown in fig. 5 to 6. It can be seen from fig. 5 to 6 that the radial deformation of the pile foundation has a great influence on the impedance function of the pile foundation-soil system, and the static stiffness of the two-dimensional pile foundation is less than that of the one-dimensional pile foundation because the side boundary of the one-dimensional pile foundation is fixed. However, the dynamic impedance peak value of the two-dimensional pile foundation is larger than that of the one-dimensional pile foundation, which means that the maximum value of the dynamic impedance of the pile foundation soil system is actually underestimated by assuming the pile foundation as the one-dimensional rod. The effect of radial deformation of the pile foundation on the pile foundation-soil system speed response is shown in fig. 7-8. It can be seen from fig. 7-8 that, for saturated soil and single-phase soil, the intensity and the position of the reflected signal of two kinds of pile foundations, namely one-dimensional pile foundation and two-dimensional pile foundation, are basically the same, and this shows that to the nondestructive test of pile foundation, the influence of the radial deformation of pile foundation is relatively less, but has great influence to the dynamic impedance of pile foundation.
According to the embodiment of the invention, the radial deformation and the vertical deformation of the pile foundation are simultaneously considered through a mathematical modeling process, the reasonability of the established axial symmetry dynamic response calculation model of the pile foundation considering the radial deformation influence is verified through a calculation diagram, and then the necessity of considering the radial deformation of the pile foundation is explained by comparing the calculation result considering the radial deformation influence with the calculation result not considering the radial deformation influence.
The above description is only for the preferred embodiment of the present invention, and is not intended to limit the scope of the present invention. Any modification, equivalent replacement, or improvement made within the spirit and principle of the present invention shall fall within the protection scope of the present invention.

Claims (6)

1. A method for determining axial symmetry dynamic response of a pile foundation considering radial deformation influence is characterized by comprising the following steps:
step S1, respectively determining radial displacement of the pile foundation caused by stress on the side surface of the pile foundation
Figure FDA0003486765660000011
Axial displacement of pile foundation caused by pile foundation side surface stress
Figure FDA0003486765660000012
Radial displacement of pile foundation caused by axial force of pile foundation bottom surface
Figure FDA0003486765660000013
Axial displacement of pile foundation caused by axial force of pile foundation bottom surface
Figure FDA0003486765660000014
Radial displacement of pile foundation caused by shear force of pile foundation bottom surface
Figure FDA0003486765660000015
Axial displacement of pile foundation caused by shear force of pile foundation bottom surface
Figure FDA0003486765660000016
Radial displacement of pile foundation caused by axial displacement of pile foundation top surface under axial load
Figure FDA0003486765660000017
Axial displacement of pile foundation caused by axial displacement of pile foundation top surface under axial load
Figure FDA0003486765660000018
Radial displacement of pile caused by stress on side surface of pile
Figure FDA0003486765660000019
And axial displacement of pile caused by stress on side surface of pile
Figure FDA00034867656600000110
Calculated by the following formula:
Figure FDA00034867656600000111
Figure FDA00034867656600000112
wherein, l represents the length of the pile foundation, and a represents the radius of the pile foundation;
Figure FDA00034867656600000113
indicating the radial displacement caused by the stress on the side surface of the pile foundation at z,
Figure FDA00034867656600000114
the axial displacement caused by the stress of the side surface of the pile foundation at the z position of the pile foundation body is represented, and z is an axial coordinate; t is tr(s) represents the radial stress of the pile base side surface at s of the pile base, s is the transverse coordinate, tz(s) axial stress of the pile base side surface at the pile base s;
Figure FDA00034867656600000115
representing radial annular loadA Green function of radial displacement at the pile foundation body (z, s) under the action of load,
Figure FDA00034867656600000116
representing the Green function of the axial displacement of the pile foundation body (z, s) under the action of radial annular load,
Figure FDA00034867656600000117
representing a Green function of radial displacement at the pile foundation body (z, s) under the action of axial annular load,
Figure FDA00034867656600000118
representing an axial displacement Green function at the pile foundation body (z, s) under the action of axial annular load; a represents the cross-sectional area of the pile foundation, and A ═ pi a2(ii) a Rho represents the density of the pile foundation, and omega represents the vibration frequency of the pile foundation;
radial displacement of pile foundation caused by axial force of pile foundation bottom surface
Figure FDA00034867656600000119
Axial displacement of pile foundation caused by axial force of pile foundation bottom surface
Figure FDA00034867656600000120
Calculated by the following formula:
Figure FDA00034867656600000121
Figure FDA00034867656600000122
Figure FDA00034867656600000123
Figure FDA0003486765660000021
Figure FDA0003486765660000022
Figure FDA0003486765660000023
wherein the content of the first and second substances,
Figure FDA0003486765660000024
the radial static displacement of the pile foundation caused by the axial force N (l) of the bottom surface of the pile foundation is shown,
Figure FDA0003486765660000025
the axial static displacement of the pile foundation caused by the axial force N (l) of the bottom surface of the pile foundation is shown,
Figure FDA0003486765660000026
represents the radial dynamic displacement of the pile foundation caused by the inertia term,
Figure FDA0003486765660000027
representing axial dynamic displacement of the pile foundation caused by the inertia term; mu is the Lame constant of the pile foundation, and upsilon is the Poisson ratio of the pile foundation;
Figure FDA0003486765660000028
the radial static displacement at the pile body s caused by the pile foundation bottom surface axial force N (l),
Figure FDA0003486765660000029
for the radial dynamic displacement at the pile body s caused by the inertia term,
Figure FDA00034867656600000210
axial static state at pile body s caused by pile foundation bottom surface axial force N (l)The displacement is carried out in such a way that,
Figure FDA00034867656600000211
axial dynamic displacement at the pile body s caused by an inertia term;
radial displacement of pile foundation caused by shear force of pile foundation bottom surface
Figure FDA00034867656600000212
Axial displacement of pile foundation caused by shear force of pile foundation bottom surface
Figure FDA00034867656600000213
Calculated by the following formula:
Figure FDA00034867656600000214
Figure FDA00034867656600000215
Figure FDA00034867656600000216
Figure FDA00034867656600000217
Figure FDA00034867656600000218
Figure FDA00034867656600000219
wherein the content of the first and second substances,
Figure FDA00034867656600000220
radial static displacement of the pile foundation caused by shearing force of the bottom surface of the pile foundation,
Figure FDA00034867656600000221
axial static displacement of the pile foundation caused by the shearing force of the bottom surface of the pile foundation,
Figure FDA00034867656600000222
for the radial dynamic displacement of the pile foundation caused by the inertia term,
Figure FDA00034867656600000223
axial dynamic displacement of the pile foundation caused by inertia;
Figure FDA00034867656600000224
j represents the polar moment of inertia of the pile foundation,
Figure FDA00034867656600000225
Figure FDA00034867656600000226
radial static displacement at the pile body s caused by the shearing force of the bottom surface of the pile,
Figure FDA00034867656600000227
for the radial dynamic displacement at the pile body s caused by the inertia term,
Figure FDA0003486765660000031
axial static displacement at the pile body s caused by the shearing force of the bottom surface of the pile,
Figure FDA0003486765660000032
axial dynamic displacement at the pile body s caused by an inertia term;
radial displacement of pile foundation caused by axial displacement of pile foundation top surface under axial load action
Figure FDA0003486765660000033
Axial displacement of pile foundation caused by axial displacement of pile foundation top surface under axial load
Figure FDA0003486765660000034
Calculated by the following formula:
Figure FDA0003486765660000035
Figure FDA0003486765660000036
Figure FDA0003486765660000037
Figure FDA0003486765660000038
Figure FDA0003486765660000039
Figure FDA00034867656600000310
wherein the content of the first and second substances,
Figure FDA00034867656600000311
is the radial static displacement of the pile foundation caused by the axial displacement of the top surface of the pile foundation under the action of the axial load,
Figure FDA00034867656600000312
the top surface of the pile foundation is under the action of axial loadAxial static displacement of the pile foundation caused by the axial displacement of the pile,
Figure FDA00034867656600000313
radial dynamic displacement of the pile foundation caused by inertia terms;
Figure FDA00034867656600000314
axial dynamic displacement of the pile foundation caused by inertia terms;
Figure FDA00034867656600000315
the radial dynamic displacement of the pile body s caused by the axial displacement of the pile top surface under the action of the axial load,
Figure FDA00034867656600000316
axial dynamic displacement at the pile body s caused by axial displacement of the top surface of the pile foundation under the action of axial load;
green function of radial displacement of pile body s under action of radial annular load
Figure FDA00034867656600000317
Green function of axial displacement of pile body s under action of radial annular load
Figure FDA00034867656600000318
Green function of radial displacement of pile body s under axial annular load
Figure FDA00034867656600000319
Green function of axial displacement of pile body s under action of axial annular load
Figure FDA00034867656600000320
Calculated by the following formula, respectively:
Figure FDA00034867656600000321
Figure FDA00034867656600000322
Figure FDA00034867656600000323
Figure FDA0003486765660000041
wherein h isj(z) is a function of the variation of the displacement of the pile in the radial direction, bj(z) is a function of the displacement of the pile in the axial direction,
Figure FDA0003486765660000042
Figure FDA0003486765660000043
is a constant of the displacement of the pile foundation when the axial annular load is 0 and the radial annular load is not 0,
Figure FDA0003486765660000044
z is more than or equal to 0 when the axial annular load is 0 and the radial annular load is not 0<The displacement constant of the pile foundation in the s area,
Figure FDA0003486765660000045
s is the axial annular load is 0 and the radial annular load is not 0<z is less than or equal to a constant of displacement of the pile foundation in the area;
Figure FDA0003486765660000046
Figure FDA0003486765660000047
is a constant of the displacement of the pile foundation when the radial annular load is 0 and the axial annular load is not 0,
Figure FDA0003486765660000048
z is more than or equal to 0 when the radial annular load is 0 and the axial annular load is not 0<The displacement constant of the pile foundation in the s area,
Figure FDA0003486765660000049
s is the radial annular load is 0 and the axial annular load is not 0<z is less than or equal to a constant of displacement of the pile foundation in the area;
step S2, according to
Figure FDA00034867656600000410
And
Figure FDA00034867656600000411
and calculating the displacement of the pile foundation by adopting the following formula:
Figure FDA00034867656600000412
wherein u isr(z) represents the radial displacement of the side surface of the pile foundation under the common action of the stress of the side surface of the pile foundation, the axial force of the bottom surface of the pile foundation, the shearing force of the bottom surface of the pile foundation and the axial displacement of the top surface of the pile foundation under the action of the axial load; u. ofzAnd (z) represents the axial displacement of the side surface of the pile foundation under the common action of the stress of the side surface of the pile foundation, the axial force of the bottom surface of the pile foundation, the shearing force of the bottom surface of the pile foundation and the axial displacement of the top surface of the pile foundation under the action of the axial load, wherein z is more than or equal to 0 and less than or equal to l, and l is the length of the pile foundation.
2. The method for determining axial symmetric dynamic response of pile foundation based on consideration of radial deformation influence according to claim 1, wherein the radial displacement of pile foundation caused by stress on side surface of pile foundation
Figure FDA00034867656600000413
Is N (l) ═ Q (l) ═ Deltaz0 and trAnd tzObtaining the radial displacement of the pile foundation when the radial displacement is not zero;
axial displacement of pile caused by stress on side surface of pile
Figure FDA00034867656600000414
Is N (l) ═ Q (l) ═ Deltaz0 and trAnd tzAxial displacement of the pile foundation is obtained when the axial displacement is not zero;
radial displacement caused by axial force of pile foundation bottom surface
Figure FDA00034867656600000415
Is Q (l) ═ Δz＝tz＝trRadial displacement of the pile foundation obtained when the displacement is 0 and N (l) is not zero;
axial displacement caused by axial force of pile foundation bottom surface
Figure FDA00034867656600000416
Is Q (l) ═ Δz＝tz＝trAxial displacement of the pile foundation obtained when the axial displacement is 0 and N (l) is not zero;
radial displacement caused by shear force of pile foundation bottom surface
Figure FDA0003486765660000051
Is N (l) ═ Δz＝tz＝trRadial displacement of the pile foundation obtained when q (l) is not zero and 0;
axial displacement caused by pile foundation bottom surface shearing force
Figure FDA0003486765660000052
Is N (l) ═ Δz＝tz＝trAxial displacement of the pile foundation obtained when q (l) is not zero and 0;
radial displacement of pile foundation caused by axial displacement of pile foundation top surface under axial load action
Figure FDA0003486765660000053
Is N (l) ═ Q (l) ═ tr＝tz0 and ΔzRadial position of pile foundation obtained when it is not zeroMoving;
axial displacement of pile foundation caused by axial displacement of pile foundation top surface under axial load action
Figure FDA0003486765660000054
Is N (l) ═ Q (l) ═ tr＝tz0 and ΔzAxial displacement of the pile foundation is obtained when the axial displacement is not zero;
wherein, trRepresenting radial stresses, t, acting on the pile-side surfacezAxial stress acting on the pile-side surface, N (l) pile-bottom axial force, Q (l) pile-bottom shear force, DeltazShowing the axial displacement of the pile top surface under axial load.
3. The method for determining the axial-symmetric dynamic response of the pile foundation considering the influence of the radial deformation as claimed in claim 1, wherein the Lame constant is E/2(1+ upsilon), and E represents the elastic modulus of the pile foundation.
4. The method for determining axial symmetric dynamic response of pile foundation based on consideration of radial deformation influence according to claim 1, wherein h is1(z)＝0，b1(z)＝1；h2(z)＝υ，
Figure FDA0003486765660000055
Figure FDA0003486765660000056
5. A method for determining axial symmetric dynamic response of pile foundation considering influence of radial deformation according to claim 1,
Figure FDA0003486765660000057
Figure FDA0003486765660000058
Figure FDA0003486765660000059
6. a method for determining axial symmetric dynamic response of pile foundation considering influence of radial deformation according to claim 1,
Figure FDA00034867656600000510
Figure FDA0003486765660000061
Figure FDA0003486765660000062
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